Step 1. Origin and Battlefield 1942

When you have installed Origin, start it and login using your details you chose during the registration process.
In Origin, enter the Store and continue to Demos. You should now be able to find Battlefield 1942, click on it and follow the instructions to install the game.

The game should now be installed in C:\Program Files\Origin Games\Battlefield 1942 or at a similar location.
Whenever you want to start Battlefield 1942 you simply click it in your MY GAMES menu and press Play.

Step 2. SilentHeroes

During the installation, choose that SilentHeroes should be installed in Origin Battlefield 1942 (C:\Program Files\Origin Games\Battlefield 1942) and NOT in the old vanilla-path (C:\Program Files\EA GAMES\Battlefield 1942).

Step 3. Custom-maps

You should now have SilentHeroes installed, but not the Custom-maps that we play during the Friday-gathering.
Follow the instructions in the Custom-maps Updater thread to install the Custom-maps.